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**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

In this paper we give fast distributed graph algorithms for detecting and listing small subgraphs, and for computing or approximating the girth. Our algorithms improve upon the state of the art by polynomial factors, and for girth, we obtain a constant-time algorithm for additive +1 approximation in Congested Clique, and the first parametrized algorithm for exact computation in Congest.
In the Congested Clique model, we first develop a technique for learning small neighborhoods, and apply it to obtain an O(1)-round algorithm that computes the girth with only an additive +1 error. Next, we introduce a new technique (the partition tree technique) allowing for efficiently listing all copies of any subgraph, which is deterministic and improves upon the state-of the-art for non-dense graphs. We give two concrete applications of the partition tree technique: First we show that for constant k, it is possible to solve C_{2k}-detection in O(1) rounds in the Congested Clique, improving on prior work, which used fast matrix multiplication and thus had polynomial round complexity. Second, we show that in triangle-free graphs, the girth can be exactly computed in time polynomially faster than the best known bounds for general graphs. We remark that no analogous result is currently known for sequential algorithms.
In the Congest model, we describe a new approach for finding cycles, and instantiate it in two ways: first, we show a fast parametrized algorithm for girth with round complexity Õ(min{g⋅ n^{1-1/Θ(g)},n}) for any girth g; and second, we show how to find small even-length cycles C_{2k} for k = 3,4,5 in O(n^{1-1/k}) rounds. This is a polynomial improvement upon the previous running times; for example, our C₆-detection algorithm runs in O(n^{2/3}) rounds, compared to O(n^{3/4}) in prior work. Finally, using our improved C₆-freeness algorithm, and the barrier on proving lower bounds on triangle-freeness of Eden et al., we show that improving the current ̃Ω(√n) lower bound for C₆-freeness of Korhonen et al. by any polynomial factor would imply strong circuit complexity lower bounds.

Keren Censor-Hillel, Orr Fischer, Tzlil Gonen, François Le Gall, Dean Leitersdorf, and Rotem Oshman. Fast Distributed Algorithms for Girth, Cycles and Small Subgraphs. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{censorhillel_et_al:LIPIcs.DISC.2020.33, author = {Censor-Hillel, Keren and Fischer, Orr and Gonen, Tzlil and Le Gall, Fran\c{c}ois and Leitersdorf, Dean and Oshman, Rotem}, title = {{Fast Distributed Algorithms for Girth, Cycles and Small Subgraphs}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {33:1--33:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.33}, URN = {urn:nbn:de:0030-drops-131115}, doi = {10.4230/LIPIcs.DISC.2020.33}, annote = {Keywords: distributed graph algorithms, cycles, girth, Congested Clique, CONGEST} }

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**Published in:** LIPIcs, Volume 95, 21st International Conference on Principles of Distributed Systems (OPODIS 2017)

In the subgraph-freeness problem, we are given a constant-sized graph H, and wish to de- termine whether the network graph contains H as a subgraph or not. Until now, the only lower bounds on subgraph-freeness known for the CONGEST model were for cycles of length greater than 3; here we extend and generalize the cycle lower bound, and obtain polynomial lower bounds for subgraph-freeness in the CONGEST model for two classes of subgraphs.
The first class contains any graph obtained by starting from a 2-connected graph H for which we already know a lower bound, and replacing the vertices of H by arbitrary connected graphs. We show that the lower bound on H carries over to the new graph. The second class is constructed by starting from a cycle Ck of length k ≥ 4, and constructing a graph H ̃ from Ck by replacing each edge {i, (i + 1) mod k} of the cycle with a connected graph Hi, subject to some constraints on the graphs H_{0}, . . . , H_{k−1}. In this case we obtain a polynomial lower bound for the new graph H ̃, depending on the size of the shortest cycle in H ̃ passing through the vertices of the original k-cycle.

Tzlil Gonen and Rotem Oshman. Lower Bounds for Subgraph Detection in the CONGEST Model. In 21st International Conference on Principles of Distributed Systems (OPODIS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 95, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gonen_et_al:LIPIcs.OPODIS.2017.6, author = {Gonen, Tzlil and Oshman, Rotem}, title = {{Lower Bounds for Subgraph Detection in the CONGEST Model}}, booktitle = {21st International Conference on Principles of Distributed Systems (OPODIS 2017)}, pages = {6:1--6:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-061-3}, ISSN = {1868-8969}, year = {2018}, volume = {95}, editor = {Aspnes, James and Bessani, Alysson and Felber, Pascal and Leit\~{a}o, Jo\~{a}o}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2017.6}, URN = {urn:nbn:de:0030-drops-86445}, doi = {10.4230/LIPIcs.OPODIS.2017.6}, annote = {Keywords: subgraph freeness, CONGEST, lower bounds} }

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**Published in:** LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

In this paper we present distributed property-testing algorithms for graph properties in the CONGEST model, with emphasis on testing subgraph-freeness. Testing a graph property P means distinguishing graphs G = (V,E) having property P from graphs that are epsilon-far from having it, meaning that epsilon|E| edges must be added or removed from G to obtain a graph satisfying P.
We present a series of results, including:
- Testing H-freeness in O(1/epsilon) rounds, for any constant-sized graph H containing an edge (u,v) such that any cycle in H contain either u or v (or both). This includes all connected graphs over five vertices except K_5. For triangles, we can do even better when epsilon is not too small.
- A deterministic CONGEST protocol determining whether a graph contains a given tree as a subgraph in constant time.
- For cliques K_s with s >= 5, we show that K_s-freeness can be tested in O(m^(1/2-1/(s-2)) epsilon^(-1/2-1/(s-2))) rounds, where m is the number of edges in the network graph.
- We describe a general procedure for converting epsilon-testers with f(D) rounds, where D denotes the diameter of the graph, to work in O((log n)/epsilon)+f((log n)/epsilon) rounds, where n is the number of processors of the network. We then apply this procedure to obtain an epsilon-tester for testing whether a graph is bipartite and testing whether a graph is cycle-free. Moreover, for cycle-freeness, we obtain a corrector of the graph that locally corrects the graph so that the corrected graph is acyclic. Note that, unlike a tester, a corrector needs to mend the graph in many places in the case that the graph is far from having the property.
These protocols extend and improve previous results of [Censor-Hillel et al. 2016] and [Fraigniaud et al. 2016].

Guy Even, Orr Fischer, Pierre Fraigniaud, Tzlil Gonen, Reut Levi, Moti Medina, Pedro Montealegre, Dennis Olivetti, Rotem Oshman, Ivan Rapaport, and Ioan Todinca. Three Notes on Distributed Property Testing. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 15:1-15:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{even_et_al:LIPIcs.DISC.2017.15, author = {Even, Guy and Fischer, Orr and Fraigniaud, Pierre and Gonen, Tzlil and Levi, Reut and Medina, Moti and Montealegre, Pedro and Olivetti, Dennis and Oshman, Rotem and Rapaport, Ivan and Todinca, Ioan}, title = {{Three Notes on Distributed Property Testing}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {15:1--15:30}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.15}, URN = {urn:nbn:de:0030-drops-79847}, doi = {10.4230/LIPIcs.DISC.2017.15}, annote = {Keywords: Property testing, Property correcting, Distributed algorithms, CONGEST model} }

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